In this section, we focus on analyzing the transient behavior of the number of jobs of each class (Theorem ).
Our analysis is motivated by the approach of fluid approximation, where an appropriately scaled system process can be approximated, as the number of servers grows large, by a deterministic system that is defined by a system of differential equations.
Such a deterministic system is referred to as fluid model in the literature.
In particular, our proof is closely related to the argument for a result called Kurtz's theorem on density-dependent Markov processes (see Chapter 8 of [1]} or Chapter \(5.3\)  of [2]}). However, we remark that there are two key differences between our proof and the standard argument. First, we scale the number of jobs of each class by the corresponding arrival rate; in contrast, traditional fluid approximation considers scaling by the number of servers. The traditional fluid scaling does not work for a system with multi-server jobs, where server needs can potentially grow with the number of servers \(n\) .
Second, because we have multiple job classes, instead of directly comparing our original system with the fluid model, we need to construct an intermediate system and couple it with our original system. We establish the fluid approximation by showing that the intermediate system is close to both the original system and the fluid model.
